{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "<p align=\"center\">\n",
    "    <img src=\"https://github.com/GeostatsGuy/GeostatsPy/blob/master/TCG_color_logo.png?raw=true\" width=\"220\" height=\"240\" />\n",
    "\n",
    "</p>\n",
    "\n",
    "## Subsurface Data Analytics \n",
    "\n",
    "## Interactive Gradient Descent\n",
    "\n",
    "#### Michael Pyrcz, Associate Professor, University of Texas at Austin \n",
    "\n",
    "##### [Twitter](https://twitter.com/geostatsguy) | [GitHub](https://github.com/GeostatsGuy) | [Website](http://michaelpyrcz.com) | [GoogleScholar](https://scholar.google.com/citations?user=QVZ20eQAAAAJ&hl=en&oi=ao) | [Book](https://www.amazon.com/Geostatistical-Reservoir-Modeling-Michael-Pyrcz/dp/0199731446) | [YouTube](https://www.youtube.com/channel/UCLqEr-xV-ceHdXXXrTId5ig)  | [LinkedIn](https://www.linkedin.com/in/michael-pyrcz-61a648a1) | [GeostatsPy](https://github.com/GeostatsGuy/GeostatsPy)\n",
    "\n",
    "\n",
    "### PGE 383 Exercise: Interactive Gradient Descent\n",
    "\n",
    "Here's a simple demonstration of gradient descent. I have a lecture that describes the basics of optimization for machine learning model training, [Optimization Lecture](https://www.youtube.com/watch?v=4nYz5j0sAQs&list=PLG19vXLQHvSC2ZKFIkgVpI9fCjkN38kwf&index=23&t=578s)\n",
    "\n",
    "In this demonstration you will be able to take a customimizable loss function.\n",
    "\n",
    "* you can vary the magnitude and spatial correlation of the noise\n",
    "\n",
    "* **magnitude of noise** is controlled by the standard deviation of a Gaussian random residual \n",
    "\n",
    "* **spatialcorrelation of noise** is controlled by the size of a Gaussian kernal applied to convolve the random noise to impose spatial structure \n",
    "\n",
    "Then you can set the:\n",
    "\n",
    "* **initial parameter combination** ($\\phi_1, \\phi_2$)\n",
    "\n",
    "* **initial learning rate** \\[0.1,5\\]\n",
    "\n",
    "* **momentum** \\[0.0,1.0\\]\n",
    "\n",
    "* **\n",
    "\n",
    "* simple polynomial model\n",
    "\n",
    "* 1 preditor feature and 1 response feature\n",
    "\n",
    "for an high interpretability model/ simple illustration.\n",
    "\n",
    "#### Train / Test Split\n",
    "\n",
    "The available data is split into training and testing subsets.\n",
    "\n",
    "* in general 15-30% of the data is withheld from training to apply as testing data\n",
    "\n",
    "* testing data selection should be fair, the same difficulty of predictions (offset/different from the training dat\n",
    "\n",
    "#### Machine Learning Model Traing\n",
    "\n",
    "The training data is applied to train the model parameters such that the model minimizes mismatch with the training data\n",
    "\n",
    "* it is common to use **mean square error** (known as a **L2 norm**) as a loss function summarizing the model mismatch\n",
    "\n",
    "* **miminizing the loss function** for simple models an anlytical solution may be available, but for most machine this requires an iterative optimization method to find the best model parameters\n",
    "\n",
    "This process is repeated over a range of model complexities specified by hyperparameters. \n",
    "\n",
    "#### Machine Learning Model Tuning\n",
    "\n",
    "The withheld testing data is retrieved and loss function (usually the **L2 norm** again) is calculated to summarize the error over the testing data\n",
    "\n",
    "* this is repeated over over the range of specified hypparameters\n",
    "\n",
    "* the model complexity / hyperparameters that minimize the loss function / error summary in testing is selected\n",
    "\n",
    "This is known are model hypparameter tuning.\n",
    "\n",
    "#### Machine Learning Model Overfit\n",
    "\n",
    "More model complexity/flexibility than can be justified with the available data, data accuracy, frequency and coverage\n",
    "\n",
    "* Model explains “idiosyncrasies” of the data, capturing data noise/error in the model\n",
    "\n",
    "* High accuracy in training, but low accuracy in testing / real-world use away from training data cases – poor ability of the model to generalize\n",
    "\n",
    "\n",
    "#### Workflow Goals\n",
    "\n",
    "Learn the basics of machine learning training, tuning for model generalization while avoiding model overfit.\n",
    "\n",
    "This includes:\n",
    "\n",
    "* Demonstrate model training and tuning by hand with an interactive exercies\n",
    "\n",
    "* Demonstrate the role of data error in leading to model overfit with complicated models\n",
    "\n",
    "#### Getting Started\n",
    "\n",
    "You will need to copy the following data files to your working directory.  They are available [here](https://github.com/GeostatsGuy/GeoDataSets):\n",
    "\n",
    "* Tabular data - [Stochastic_1D_por_perm_demo.csv](https://github.com/GeostatsGuy/GeoDataSets/blob/master/Stochastic_1D_por_perm_demo.csv)\n",
    "* Tabular data - [Random_Parabola.csv](https://github.com/GeostatsGuy/GeoDataSets/blob/master/Random_Parabola.csv)\n",
    "\n",
    "These datasets are available in the folder: https://github.com/GeostatsGuy/GeoDataSets.\n",
    "\n",
    "\n",
    "#### Import Required Packages\n",
    "\n",
    "We will also need some standard packages. These should have been installed with Anaconda 3."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "%matplotlib notebook\n",
    "import sys                                              # supress output to screen for interactive variogram modeling                                             \n",
    "import math                                             # square root operator\n",
    "import numpy as np                                      # arrays and matrix math\n",
    "import matplotlib.pyplot as plt                         # plotting\n",
    "import scipy                                            # kernel for convolution\n",
    "from ipywidgets import interactive                      # widgets and interactivity\n",
    "from ipywidgets import widgets                            \n",
    "from ipywidgets import Layout\n",
    "from ipywidgets import Label\n",
    "from ipywidgets import VBox, HBox\n",
    "from scipy.interpolate import griddata\n",
    "from matplotlib.animation import FuncAnimation\n",
    "import scipy.signal as signal\n",
    "from scipy.interpolate import RegularGridInterpolator\n",
    "from matplotlib import animation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Animated Optimization\n",
    "\n",
    "\n",
    "\n",
    "#### Make the Exhaustive Loss Function\n",
    "\n",
    "Let's ensure a global minimum in the middle of the solution space and add some structured noise to increase the difficulty\n",
    "\n",
    "* local minimums for gradient descent to get stuck"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "nx = 100; ny = 100                                        # size of the loss function grid\n",
    "cmap = plt.cm.inferno                                     # set the colormap\n",
    "xmin = 0.0; xmax = 100.0; ymin = 0.0; ymax = 100.0        # set the extents\n",
    "X, Y = np.mgrid[ymin:ymax:complex(ny), 0:100:complex(nx)]\n",
    "XY = np.vstack((Y.flatten(), X.flatten())).T              # make location list for interpolation method\n",
    "XY_list = list(map(tuple, XY))\n",
    "x = np.zeros(100); y = np.zeros(100)                    "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
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       "/* Put everything inside the global mpl namespace */\n",
       "window.mpl = {};\n",
       "\n",
       "\n",
       "mpl.get_websocket_type = function() {\n",
       "    if (typeof(WebSocket) !== 'undefined') {\n",
       "        return WebSocket;\n",
       "    } else if (typeof(MozWebSocket) !== 'undefined') {\n",
       "        return MozWebSocket;\n",
       "    } else {\n",
       "        alert('Your browser does not have WebSocket support. ' +\n",
       "              'Please try Chrome, Safari or Firefox ≥ 6. ' +\n",
       "              'Firefox 4 and 5 are also supported but you ' +\n",
       "              'have to enable WebSockets in about:config.');\n",
       "    };\n",
       "}\n",
       "\n",
       "mpl.figure = function(figure_id, websocket, ondownload, parent_element) {\n",
       "    this.id = figure_id;\n",
       "\n",
       "    this.ws = websocket;\n",
       "\n",
       "    this.supports_binary = (this.ws.binaryType != undefined);\n",
       "\n",
       "    if (!this.supports_binary) {\n",
       "        var warnings = document.getElementById(\"mpl-warnings\");\n",
       "        if (warnings) {\n",
       "            warnings.style.display = 'block';\n",
       "            warnings.textContent = (\n",
       "                \"This browser does not support binary websocket messages. \" +\n",
       "                    \"Performance may be slow.\");\n",
       "        }\n",
       "    }\n",
       "\n",
       "    this.imageObj = new Image();\n",
       "\n",
       "    this.context = undefined;\n",
       "    this.message = undefined;\n",
       "    this.canvas = undefined;\n",
       "    this.rubberband_canvas = undefined;\n",
       "    this.rubberband_context = undefined;\n",
       "    this.format_dropdown = undefined;\n",
       "\n",
       "    this.image_mode = 'full';\n",
       "\n",
       "    this.root = $('<div/>');\n",
       "    this._root_extra_style(this.root)\n",
       "    this.root.attr('style', 'display: inline-block');\n",
       "\n",
       "    $(parent_element).append(this.root);\n",
       "\n",
       "    this._init_header(this);\n",
       "    this._init_canvas(this);\n",
       "    this._init_toolbar(this);\n",
       "\n",
       "    var fig = this;\n",
       "\n",
       "    this.waiting = false;\n",
       "\n",
       "    this.ws.onopen =  function () {\n",
       "            fig.send_message(\"supports_binary\", {value: fig.supports_binary});\n",
       "            fig.send_message(\"send_image_mode\", {});\n",
       "            if (mpl.ratio != 1) {\n",
       "                fig.send_message(\"set_dpi_ratio\", {'dpi_ratio': mpl.ratio});\n",
       "            }\n",
       "            fig.send_message(\"refresh\", {});\n",
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       "\n",
       "    this.imageObj.onload = function() {\n",
       "            if (fig.image_mode == 'full') {\n",
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       "            fig.context.drawImage(fig.imageObj, 0, 0);\n",
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       "    this.imageObj.onunload = function() {\n",
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       "\n",
       "    this.ws.onmessage = this._make_on_message_function(this);\n",
       "\n",
       "    this.ondownload = ondownload;\n",
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       "\n",
       "mpl.figure.prototype._init_header = function() {\n",
       "    var titlebar = $(\n",
       "        '<div class=\"ui-dialog-titlebar ui-widget-header ui-corner-all ' +\n",
       "        'ui-helper-clearfix\"/>');\n",
       "    var titletext = $(\n",
       "        '<div class=\"ui-dialog-title\" style=\"width: 100%; ' +\n",
       "        'text-align: center; padding: 3px;\"/>');\n",
       "    titlebar.append(titletext)\n",
       "    this.root.append(titlebar);\n",
       "    this.header = titletext[0];\n",
       "}\n",
       "\n",
       "\n",
       "\n",
       "mpl.figure.prototype._canvas_extra_style = function(canvas_div) {\n",
       "\n",
       "}\n",
       "\n",
       "\n",
       "mpl.figure.prototype._root_extra_style = function(canvas_div) {\n",
       "\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._init_canvas = function() {\n",
       "    var fig = this;\n",
       "\n",
       "    var canvas_div = $('<div/>');\n",
       "\n",
       "    canvas_div.attr('style', 'position: relative; clear: both; outline: 0');\n",
       "\n",
       "    function canvas_keyboard_event(event) {\n",
       "        return fig.key_event(event, event['data']);\n",
       "    }\n",
       "\n",
       "    canvas_div.keydown('key_press', canvas_keyboard_event);\n",
       "    canvas_div.keyup('key_release', canvas_keyboard_event);\n",
       "    this.canvas_div = canvas_div\n",
       "    this._canvas_extra_style(canvas_div)\n",
       "    this.root.append(canvas_div);\n",
       "\n",
       "    var canvas = $('<canvas/>');\n",
       "    canvas.addClass('mpl-canvas');\n",
       "    canvas.attr('style', \"left: 0; top: 0; z-index: 0; outline: 0\")\n",
       "\n",
       "    this.canvas = canvas[0];\n",
       "    this.context = canvas[0].getContext(\"2d\");\n",
       "\n",
       "    var backingStore = this.context.backingStorePixelRatio ||\n",
       "\tthis.context.webkitBackingStorePixelRatio ||\n",
       "\tthis.context.mozBackingStorePixelRatio ||\n",
       "\tthis.context.msBackingStorePixelRatio ||\n",
       "\tthis.context.oBackingStorePixelRatio ||\n",
       "\tthis.context.backingStorePixelRatio || 1;\n",
       "\n",
       "    mpl.ratio = (window.devicePixelRatio || 1) / backingStore;\n",
       "\n",
       "    var rubberband = $('<canvas/>');\n",
       "    rubberband.attr('style', \"position: absolute; left: 0; top: 0; z-index: 1;\")\n",
       "\n",
       "    var pass_mouse_events = true;\n",
       "\n",
       "    canvas_div.resizable({\n",
       "        start: function(event, ui) {\n",
       "            pass_mouse_events = false;\n",
       "        },\n",
       "        resize: function(event, ui) {\n",
       "            fig.request_resize(ui.size.width, ui.size.height);\n",
       "        },\n",
       "        stop: function(event, ui) {\n",
       "            pass_mouse_events = true;\n",
       "            fig.request_resize(ui.size.width, ui.size.height);\n",
       "        },\n",
       "    });\n",
       "\n",
       "    function mouse_event_fn(event) {\n",
       "        if (pass_mouse_events)\n",
       "            return fig.mouse_event(event, event['data']);\n",
       "    }\n",
       "\n",
       "    rubberband.mousedown('button_press', mouse_event_fn);\n",
       "    rubberband.mouseup('button_release', mouse_event_fn);\n",
       "    // Throttle sequential mouse events to 1 every 20ms.\n",
       "    rubberband.mousemove('motion_notify', mouse_event_fn);\n",
       "\n",
       "    rubberband.mouseenter('figure_enter', mouse_event_fn);\n",
       "    rubberband.mouseleave('figure_leave', mouse_event_fn);\n",
       "\n",
       "    canvas_div.on(\"wheel\", function (event) {\n",
       "        event = event.originalEvent;\n",
       "        event['data'] = 'scroll'\n",
       "        if (event.deltaY < 0) {\n",
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       "            event.step = -1;\n",
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       "        mouse_event_fn(event);\n",
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       "\n",
       "    canvas_div.append(canvas);\n",
       "    canvas_div.append(rubberband);\n",
       "\n",
       "    this.rubberband = rubberband;\n",
       "    this.rubberband_canvas = rubberband[0];\n",
       "    this.rubberband_context = rubberband[0].getContext(\"2d\");\n",
       "    this.rubberband_context.strokeStyle = \"#000000\";\n",
       "\n",
       "    this._resize_canvas = function(width, height) {\n",
       "        // Keep the size of the canvas, canvas container, and rubber band\n",
       "        // canvas in synch.\n",
       "        canvas_div.css('width', width)\n",
       "        canvas_div.css('height', height)\n",
       "\n",
       "        canvas.attr('width', width * mpl.ratio);\n",
       "        canvas.attr('height', height * mpl.ratio);\n",
       "        canvas.attr('style', 'width: ' + width + 'px; height: ' + height + 'px;');\n",
       "\n",
       "        rubberband.attr('width', width);\n",
       "        rubberband.attr('height', height);\n",
       "    }\n",
       "\n",
       "    // Set the figure to an initial 600x600px, this will subsequently be updated\n",
       "    // upon first draw.\n",
       "    this._resize_canvas(600, 600);\n",
       "\n",
       "    // Disable right mouse context menu.\n",
       "    $(this.rubberband_canvas).bind(\"contextmenu\",function(e){\n",
       "        return false;\n",
       "    });\n",
       "\n",
       "    function set_focus () {\n",
       "        canvas.focus();\n",
       "        canvas_div.focus();\n",
       "    }\n",
       "\n",
       "    window.setTimeout(set_focus, 100);\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._init_toolbar = function() {\n",
       "    var fig = this;\n",
       "\n",
       "    var nav_element = $('<div/>');\n",
       "    nav_element.attr('style', 'width: 100%');\n",
       "    this.root.append(nav_element);\n",
       "\n",
       "    // Define a callback function for later on.\n",
       "    function toolbar_event(event) {\n",
       "        return fig.toolbar_button_onclick(event['data']);\n",
       "    }\n",
       "    function toolbar_mouse_event(event) {\n",
       "        return fig.toolbar_button_onmouseover(event['data']);\n",
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       "\n",
       "    for(var toolbar_ind in mpl.toolbar_items) {\n",
       "        var name = mpl.toolbar_items[toolbar_ind][0];\n",
       "        var tooltip = mpl.toolbar_items[toolbar_ind][1];\n",
       "        var image = mpl.toolbar_items[toolbar_ind][2];\n",
       "        var method_name = mpl.toolbar_items[toolbar_ind][3];\n",
       "\n",
       "        if (!name) {\n",
       "            // put a spacer in here.\n",
       "            continue;\n",
       "        }\n",
       "        var button = $('<button/>');\n",
       "        button.addClass('ui-button ui-widget ui-state-default ui-corner-all ' +\n",
       "                        'ui-button-icon-only');\n",
       "        button.attr('role', 'button');\n",
       "        button.attr('aria-disabled', 'false');\n",
       "        button.click(method_name, toolbar_event);\n",
       "        button.mouseover(tooltip, toolbar_mouse_event);\n",
       "\n",
       "        var icon_img = $('<span/>');\n",
       "        icon_img.addClass('ui-button-icon-primary ui-icon');\n",
       "        icon_img.addClass(image);\n",
       "        icon_img.addClass('ui-corner-all');\n",
       "\n",
       "        var tooltip_span = $('<span/>');\n",
       "        tooltip_span.addClass('ui-button-text');\n",
       "        tooltip_span.html(tooltip);\n",
       "\n",
       "        button.append(icon_img);\n",
       "        button.append(tooltip_span);\n",
       "\n",
       "        nav_element.append(button);\n",
       "    }\n",
       "\n",
       "    var fmt_picker_span = $('<span/>');\n",
       "\n",
       "    var fmt_picker = $('<select/>');\n",
       "    fmt_picker.addClass('mpl-toolbar-option ui-widget ui-widget-content');\n",
       "    fmt_picker_span.append(fmt_picker);\n",
       "    nav_element.append(fmt_picker_span);\n",
       "    this.format_dropdown = fmt_picker[0];\n",
       "\n",
       "    for (var ind in mpl.extensions) {\n",
       "        var fmt = mpl.extensions[ind];\n",
       "        var option = $(\n",
       "            '<option/>', {selected: fmt === mpl.default_extension}).html(fmt);\n",
       "        fmt_picker.append(option);\n",
       "    }\n",
       "\n",
       "    // Add hover states to the ui-buttons\n",
       "    $( \".ui-button\" ).hover(\n",
       "        function() { $(this).addClass(\"ui-state-hover\");},\n",
       "        function() { $(this).removeClass(\"ui-state-hover\");}\n",
       "    );\n",
       "\n",
       "    var status_bar = $('<span class=\"mpl-message\"/>');\n",
       "    nav_element.append(status_bar);\n",
       "    this.message = status_bar[0];\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.request_resize = function(x_pixels, y_pixels) {\n",
       "    // Request matplotlib to resize the figure. Matplotlib will then trigger a resize in the client,\n",
       "    // which will in turn request a refresh of the image.\n",
       "    this.send_message('resize', {'width': x_pixels, 'height': y_pixels});\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.send_message = function(type, properties) {\n",
       "    properties['type'] = type;\n",
       "    properties['figure_id'] = this.id;\n",
       "    this.ws.send(JSON.stringify(properties));\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.send_draw_message = function() {\n",
       "    if (!this.waiting) {\n",
       "        this.waiting = true;\n",
       "        this.ws.send(JSON.stringify({type: \"draw\", figure_id: this.id}));\n",
       "    }\n",
       "}\n",
       "\n",
       "\n",
       "mpl.figure.prototype.handle_save = function(fig, msg) {\n",
       "    var format_dropdown = fig.format_dropdown;\n",
       "    var format = format_dropdown.options[format_dropdown.selectedIndex].value;\n",
       "    fig.ondownload(fig, format);\n",
       "}\n",
       "\n",
       "\n",
       "mpl.figure.prototype.handle_resize = function(fig, msg) {\n",
       "    var size = msg['size'];\n",
       "    if (size[0] != fig.canvas.width || size[1] != fig.canvas.height) {\n",
       "        fig._resize_canvas(size[0], size[1]);\n",
       "        fig.send_message(\"refresh\", {});\n",
       "    };\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_rubberband = function(fig, msg) {\n",
       "    var x0 = msg['x0'] / mpl.ratio;\n",
       "    var y0 = (fig.canvas.height - msg['y0']) / mpl.ratio;\n",
       "    var x1 = msg['x1'] / mpl.ratio;\n",
       "    var y1 = (fig.canvas.height - msg['y1']) / mpl.ratio;\n",
       "    x0 = Math.floor(x0) + 0.5;\n",
       "    y0 = Math.floor(y0) + 0.5;\n",
       "    x1 = Math.floor(x1) + 0.5;\n",
       "    y1 = Math.floor(y1) + 0.5;\n",
       "    var min_x = Math.min(x0, x1);\n",
       "    var min_y = Math.min(y0, y1);\n",
       "    var width = Math.abs(x1 - x0);\n",
       "    var height = Math.abs(y1 - y0);\n",
       "\n",
       "    fig.rubberband_context.clearRect(\n",
       "        0, 0, fig.canvas.width / mpl.ratio, fig.canvas.height / mpl.ratio);\n",
       "\n",
       "    fig.rubberband_context.strokeRect(min_x, min_y, width, height);\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_figure_label = function(fig, msg) {\n",
       "    // Updates the figure title.\n",
       "    fig.header.textContent = msg['label'];\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_cursor = function(fig, msg) {\n",
       "    var cursor = msg['cursor'];\n",
       "    switch(cursor)\n",
       "    {\n",
       "    case 0:\n",
       "        cursor = 'pointer';\n",
       "        break;\n",
       "    case 1:\n",
       "        cursor = 'default';\n",
       "        break;\n",
       "    case 2:\n",
       "        cursor = 'crosshair';\n",
       "        break;\n",
       "    case 3:\n",
       "        cursor = 'move';\n",
       "        break;\n",
       "    }\n",
       "    fig.rubberband_canvas.style.cursor = cursor;\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_message = function(fig, msg) {\n",
       "    fig.message.textContent = msg['message'];\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_draw = function(fig, msg) {\n",
       "    // Request the server to send over a new figure.\n",
       "    fig.send_draw_message();\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_image_mode = function(fig, msg) {\n",
       "    fig.image_mode = msg['mode'];\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.updated_canvas_event = function() {\n",
       "    // Called whenever the canvas gets updated.\n",
       "    this.send_message(\"ack\", {});\n",
       "}\n",
       "\n",
       "// A function to construct a web socket function for onmessage handling.\n",
       "// Called in the figure constructor.\n",
       "mpl.figure.prototype._make_on_message_function = function(fig) {\n",
       "    return function socket_on_message(evt) {\n",
       "        if (evt.data instanceof Blob) {\n",
       "            /* FIXME: We get \"Resource interpreted as Image but\n",
       "             * transferred with MIME type text/plain:\" errors on\n",
       "             * Chrome.  But how to set the MIME type?  It doesn't seem\n",
       "             * to be part of the websocket stream */\n",
       "            evt.data.type = \"image/png\";\n",
       "\n",
       "            /* Free the memory for the previous frames */\n",
       "            if (fig.imageObj.src) {\n",
       "                (window.URL || window.webkitURL).revokeObjectURL(\n",
       "                    fig.imageObj.src);\n",
       "            }\n",
       "\n",
       "            fig.imageObj.src = (window.URL || window.webkitURL).createObjectURL(\n",
       "                evt.data);\n",
       "            fig.updated_canvas_event();\n",
       "            fig.waiting = false;\n",
       "            return;\n",
       "        }\n",
       "        else if (typeof evt.data === 'string' && evt.data.slice(0, 21) == \"data:image/png;base64\") {\n",
       "            fig.imageObj.src = evt.data;\n",
       "            fig.updated_canvas_event();\n",
       "            fig.waiting = false;\n",
       "            return;\n",
       "        }\n",
       "\n",
       "        var msg = JSON.parse(evt.data);\n",
       "        var msg_type = msg['type'];\n",
       "\n",
       "        // Call the  \"handle_{type}\" callback, which takes\n",
       "        // the figure and JSON message as its only arguments.\n",
       "        try {\n",
       "            var callback = fig[\"handle_\" + msg_type];\n",
       "        } catch (e) {\n",
       "            console.log(\"No handler for the '\" + msg_type + \"' message type: \", msg);\n",
       "            return;\n",
       "        }\n",
       "\n",
       "        if (callback) {\n",
       "            try {\n",
       "                // console.log(\"Handling '\" + msg_type + \"' message: \", msg);\n",
       "                callback(fig, msg);\n",
       "            } catch (e) {\n",
       "                console.log(\"Exception inside the 'handler_\" + msg_type + \"' callback:\", e, e.stack, msg);\n",
       "            }\n",
       "        }\n",
       "    };\n",
       "}\n",
       "\n",
       "// from http://stackoverflow.com/questions/1114465/getting-mouse-location-in-canvas\n",
       "mpl.findpos = function(e) {\n",
       "    //this section is from http://www.quirksmode.org/js/events_properties.html\n",
       "    var targ;\n",
       "    if (!e)\n",
       "        e = window.event;\n",
       "    if (e.target)\n",
       "        targ = e.target;\n",
       "    else if (e.srcElement)\n",
       "        targ = e.srcElement;\n",
       "    if (targ.nodeType == 3) // defeat Safari bug\n",
       "        targ = targ.parentNode;\n",
       "\n",
       "    // jQuery normalizes the pageX and pageY\n",
       "    // pageX,Y are the mouse positions relative to the document\n",
       "    // offset() returns the position of the element relative to the document\n",
       "    var x = e.pageX - $(targ).offset().left;\n",
       "    var y = e.pageY - $(targ).offset().top;\n",
       "\n",
       "    return {\"x\": x, \"y\": y};\n",
       "};\n",
       "\n",
       "/*\n",
       " * return a copy of an object with only non-object keys\n",
       " * we need this to avoid circular references\n",
       " * http://stackoverflow.com/a/24161582/3208463\n",
       " */\n",
       "function simpleKeys (original) {\n",
       "  return Object.keys(original).reduce(function (obj, key) {\n",
       "    if (typeof original[key] !== 'object')\n",
       "        obj[key] = original[key]\n",
       "    return obj;\n",
       "  }, {});\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.mouse_event = function(event, name) {\n",
       "    var canvas_pos = mpl.findpos(event)\n",
       "\n",
       "    if (name === 'button_press')\n",
       "    {\n",
       "        this.canvas.focus();\n",
       "        this.canvas_div.focus();\n",
       "    }\n",
       "\n",
       "    var x = canvas_pos.x * mpl.ratio;\n",
       "    var y = canvas_pos.y * mpl.ratio;\n",
       "\n",
       "    this.send_message(name, {x: x, y: y, button: event.button,\n",
       "                             step: event.step,\n",
       "                             guiEvent: simpleKeys(event)});\n",
       "\n",
       "    /* This prevents the web browser from automatically changing to\n",
       "     * the text insertion cursor when the button is pressed.  We want\n",
       "     * to control all of the cursor setting manually through the\n",
       "     * 'cursor' event from matplotlib */\n",
       "    event.preventDefault();\n",
       "    return false;\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._key_event_extra = function(event, name) {\n",
       "    // Handle any extra behaviour associated with a key event\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.key_event = function(event, name) {\n",
       "\n",
       "    // Prevent repeat events\n",
       "    if (name == 'key_press')\n",
       "    {\n",
       "        if (event.which === this._key)\n",
       "            return;\n",
       "        else\n",
       "            this._key = event.which;\n",
       "    }\n",
       "    if (name == 'key_release')\n",
       "        this._key = null;\n",
       "\n",
       "    var value = '';\n",
       "    if (event.ctrlKey && event.which != 17)\n",
       "        value += \"ctrl+\";\n",
       "    if (event.altKey && event.which != 18)\n",
       "        value += \"alt+\";\n",
       "    if (event.shiftKey && event.which != 16)\n",
       "        value += \"shift+\";\n",
       "\n",
       "    value += 'k';\n",
       "    value += event.which.toString();\n",
       "\n",
       "    this._key_event_extra(event, name);\n",
       "\n",
       "    this.send_message(name, {key: value,\n",
       "                             guiEvent: simpleKeys(event)});\n",
       "    return false;\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.toolbar_button_onclick = function(name) {\n",
       "    if (name == 'download') {\n",
       "        this.handle_save(this, null);\n",
       "    } else {\n",
       "        this.send_message(\"toolbar_button\", {name: name});\n",
       "    }\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.toolbar_button_onmouseover = function(tooltip) {\n",
       "    this.message.textContent = tooltip;\n",
       "};\n",
       "mpl.toolbar_items = [[\"Home\", \"Reset original view\", \"fa fa-home icon-home\", \"home\"], [\"Back\", \"Back to previous view\", \"fa fa-arrow-left icon-arrow-left\", \"back\"], [\"Forward\", \"Forward to next view\", \"fa fa-arrow-right icon-arrow-right\", \"forward\"], [\"\", \"\", \"\", \"\"], [\"Pan\", \"Pan axes with left mouse, zoom with right\", \"fa fa-arrows icon-move\", \"pan\"], [\"Zoom\", \"Zoom to rectangle\", \"fa fa-square-o icon-check-empty\", \"zoom\"], [\"\", \"\", \"\", \"\"], [\"Download\", \"Download plot\", \"fa fa-floppy-o icon-save\", \"download\"]];\n",
       "\n",
       "mpl.extensions = [\"eps\", \"jpeg\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\", \"tif\"];\n",
       "\n",
       "mpl.default_extension = \"png\";var comm_websocket_adapter = function(comm) {\n",
       "    // Create a \"websocket\"-like object which calls the given IPython comm\n",
       "    // object with the appropriate methods. Currently this is a non binary\n",
       "    // socket, so there is still some room for performance tuning.\n",
       "    var ws = {};\n",
       "\n",
       "    ws.close = function() {\n",
       "        comm.close()\n",
       "    };\n",
       "    ws.send = function(m) {\n",
       "        //console.log('sending', m);\n",
       "        comm.send(m);\n",
       "    };\n",
       "    // Register the callback with on_msg.\n",
       "    comm.on_msg(function(msg) {\n",
       "        //console.log('receiving', msg['content']['data'], msg);\n",
       "        // Pass the mpl event to the overridden (by mpl) onmessage function.\n",
       "        ws.onmessage(msg['content']['data'])\n",
       "    });\n",
       "    return ws;\n",
       "}\n",
       "\n",
       "mpl.mpl_figure_comm = function(comm, msg) {\n",
       "    // This is the function which gets called when the mpl process\n",
       "    // starts-up an IPython Comm through the \"matplotlib\" channel.\n",
       "\n",
       "    var id = msg.content.data.id;\n",
       "    // Get hold of the div created by the display call when the Comm\n",
       "    // socket was opened in Python.\n",
       "    var element = $(\"#\" + id);\n",
       "    var ws_proxy = comm_websocket_adapter(comm)\n",
       "\n",
       "    function ondownload(figure, format) {\n",
       "        window.open(figure.imageObj.src);\n",
       "    }\n",
       "\n",
       "    var fig = new mpl.figure(id, ws_proxy,\n",
       "                           ondownload,\n",
       "                           element.get(0));\n",
       "\n",
       "    // Call onopen now - mpl needs it, as it is assuming we've passed it a real\n",
       "    // web socket which is closed, not our websocket->open comm proxy.\n",
       "    ws_proxy.onopen();\n",
       "\n",
       "    fig.parent_element = element.get(0);\n",
       "    fig.cell_info = mpl.find_output_cell(\"<div id='\" + id + \"'></div>\");\n",
       "    if (!fig.cell_info) {\n",
       "        console.error(\"Failed to find cell for figure\", id, fig);\n",
       "        return;\n",
       "    }\n",
       "\n",
       "    var output_index = fig.cell_info[2]\n",
       "    var cell = fig.cell_info[0];\n",
       "\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.handle_close = function(fig, msg) {\n",
       "    var width = fig.canvas.width/mpl.ratio\n",
       "    fig.root.unbind('remove')\n",
       "\n",
       "    // Update the output cell to use the data from the current canvas.\n",
       "    fig.push_to_output();\n",
       "    var dataURL = fig.canvas.toDataURL();\n",
       "    // Re-enable the keyboard manager in IPython - without this line, in FF,\n",
       "    // the notebook keyboard shortcuts fail.\n",
       "    IPython.keyboard_manager.enable()\n",
       "    $(fig.parent_element).html('<img src=\"' + dataURL + '\" width=\"' + width + '\">');\n",
       "    fig.close_ws(fig, msg);\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.close_ws = function(fig, msg){\n",
       "    fig.send_message('closing', msg);\n",
       "    // fig.ws.close()\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.push_to_output = function(remove_interactive) {\n",
       "    // Turn the data on the canvas into data in the output cell.\n",
       "    var width = this.canvas.width/mpl.ratio\n",
       "    var dataURL = this.canvas.toDataURL();\n",
       "    this.cell_info[1]['text/html'] = '<img src=\"' + dataURL + '\" width=\"' + width + '\">';\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.updated_canvas_event = function() {\n",
       "    // Tell IPython that the notebook contents must change.\n",
       "    IPython.notebook.set_dirty(true);\n",
       "    this.send_message(\"ack\", {});\n",
       "    var fig = this;\n",
       "    // Wait a second, then push the new image to the DOM so\n",
       "    // that it is saved nicely (might be nice to debounce this).\n",
       "    setTimeout(function () { fig.push_to_output() }, 1000);\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._init_toolbar = function() {\n",
       "    var fig = this;\n",
       "\n",
       "    var nav_element = $('<div/>');\n",
       "    nav_element.attr('style', 'width: 100%');\n",
       "    this.root.append(nav_element);\n",
       "\n",
       "    // Define a callback function for later on.\n",
       "    function toolbar_event(event) {\n",
       "        return fig.toolbar_button_onclick(event['data']);\n",
       "    }\n",
       "    function toolbar_mouse_event(event) {\n",
       "        return fig.toolbar_button_onmouseover(event['data']);\n",
       "    }\n",
       "\n",
       "    for(var toolbar_ind in mpl.toolbar_items){\n",
       "        var name = mpl.toolbar_items[toolbar_ind][0];\n",
       "        var tooltip = mpl.toolbar_items[toolbar_ind][1];\n",
       "        var image = mpl.toolbar_items[toolbar_ind][2];\n",
       "        var method_name = mpl.toolbar_items[toolbar_ind][3];\n",
       "\n",
       "        if (!name) { continue; };\n",
       "\n",
       "        var button = $('<button class=\"btn btn-default\" href=\"#\" title=\"' + name + '\"><i class=\"fa ' + image + ' fa-lg\"></i></button>');\n",
       "        button.click(method_name, toolbar_event);\n",
       "        button.mouseover(tooltip, toolbar_mouse_event);\n",
       "        nav_element.append(button);\n",
       "    }\n",
       "\n",
       "    // Add the status bar.\n",
       "    var status_bar = $('<span class=\"mpl-message\" style=\"text-align:right; float: right;\"/>');\n",
       "    nav_element.append(status_bar);\n",
       "    this.message = status_bar[0];\n",
       "\n",
       "    // Add the close button to the window.\n",
       "    var buttongrp = $('<div class=\"btn-group inline pull-right\"></div>');\n",
       "    var button = $('<button class=\"btn btn-mini btn-primary\" href=\"#\" title=\"Stop Interaction\"><i class=\"fa fa-power-off icon-remove icon-large\"></i></button>');\n",
       "    button.click(function (evt) { fig.handle_close(fig, {}); } );\n",
       "    button.mouseover('Stop Interaction', toolbar_mouse_event);\n",
       "    buttongrp.append(button);\n",
       "    var titlebar = this.root.find($('.ui-dialog-titlebar'));\n",
       "    titlebar.prepend(buttongrp);\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._root_extra_style = function(el){\n",
       "    var fig = this\n",
       "    el.on(\"remove\", function(){\n",
       "\tfig.close_ws(fig, {});\n",
       "    });\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._canvas_extra_style = function(el){\n",
       "    // this is important to make the div 'focusable\n",
       "    el.attr('tabindex', 0)\n",
       "    // reach out to IPython and tell the keyboard manager to turn it's self\n",
       "    // off when our div gets focus\n",
       "\n",
       "    // location in version 3\n",
       "    if (IPython.notebook.keyboard_manager) {\n",
       "        IPython.notebook.keyboard_manager.register_events(el);\n",
       "    }\n",
       "    else {\n",
       "        // location in version 2\n",
       "        IPython.keyboard_manager.register_events(el);\n",
       "    }\n",
       "\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._key_event_extra = function(event, name) {\n",
       "    var manager = IPython.notebook.keyboard_manager;\n",
       "    if (!manager)\n",
       "        manager = IPython.keyboard_manager;\n",
       "\n",
       "    // Check for shift+enter\n",
       "    if (event.shiftKey && event.which == 13) {\n",
       "        this.canvas_div.blur();\n",
       "        // select the cell after this one\n",
       "        var index = IPython.notebook.find_cell_index(this.cell_info[0]);\n",
       "        IPython.notebook.select(index + 1);\n",
       "    }\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_save = function(fig, msg) {\n",
       "    fig.ondownload(fig, null);\n",
       "}\n",
       "\n",
       "\n",
       "mpl.find_output_cell = function(html_output) {\n",
       "    // Return the cell and output element which can be found *uniquely* in the notebook.\n",
       "    // Note - this is a bit hacky, but it is done because the \"notebook_saving.Notebook\"\n",
       "    // IPython event is triggered only after the cells have been serialised, which for\n",
       "    // our purposes (turning an active figure into a static one), is too late.\n",
       "    var cells = IPython.notebook.get_cells();\n",
       "    var ncells = cells.length;\n",
       "    for (var i=0; i<ncells; i++) {\n",
       "        var cell = cells[i];\n",
       "        if (cell.cell_type === 'code'){\n",
       "            for (var j=0; j<cell.output_area.outputs.length; j++) {\n",
       "                var data = cell.output_area.outputs[j];\n",
       "                if (data.data) {\n",
       "                    // IPython >= 3 moved mimebundle to data attribute of output\n",
       "                    data = data.data;\n",
       "                }\n",
       "                if (data['text/html'] == html_output) {\n",
       "                    return [cell, data, j];\n",
       "                }\n",
       "            }\n",
       "        }\n",
       "    }\n",
       "}\n",
       "\n",
       "// Register the function which deals with the matplotlib target/channel.\n",
       "// The kernel may be null if the page has been refreshed.\n",
       "if (IPython.notebook.kernel != null) {\n",
       "    IPython.notebook.kernel.comm_manager.register_target('matplotlib', mpl.mpl_figure_comm);\n",
       "}\n"
      ],
      "text/plain": [
       "<IPython.core.display.Javascript object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<img src=\"data:image/png;base64,[encoded data, 42104 characters]\" width=\"719.1333333333333\">"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "Text(0.5, 1.0, 'Gradient Descent')"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Make the truth loss function\n",
    "nsmooth = 20\n",
    "np.random.seed(73073)                                # set random number seed for repeatable results\n",
    "\n",
    "truth = np.zeros([100,100])\n",
    "for iy in range(100):\n",
    "    for ix in range(100):\n",
    "        truth[iy,ix] = math.sqrt((iy-50)*(iy-50) + (ix-50)*(ix-50))\n",
    "\n",
    "noise = np.random.normal(loc=0.0, scale=5.0, size=(100,100))\n",
    "w = np.ones((nsmooth,nsmooth))/(nsmooth*nsmooth)\n",
    "\n",
    "gkern1d = signal.gaussian(nsmooth, std=nsmooth*0.2).reshape(nsmooth, 1); gkern2d = np.outer(gkern1d, gkern1d) # Teddy Hartanto, 2017, stack overflow \n",
    "gkern2d = gkern2d/np.sum(gkern2d.flatten())\n",
    "\n",
    "noise = signal.convolve2d(noise,gkern2d,mode = 'same',boundary='symm')   \n",
    "\n",
    "truth = truth + noise*10\n",
    "\n",
    "truth = (truth - np.min(truth))/(np.max(truth) - np.min(truth))*100.0\n",
    "\n",
    "# initialize the fast interpolator to evaluate the loss function at any location\n",
    "interp = RegularGridInterpolator((np.linspace(ymin,ymax,ny),np.linspace(xmin,xmax,nx)),truth, method = 'linear')\n",
    "x = np.zeros(100); y = np.zeros(100)                          # descent path arrays\n",
    "\n",
    "fig = plt.figure(figsize = (10,10)); ax = plt.gca()\n",
    "plt.xlim([0,100]); plt.ylim([0,100])\n",
    "c1 = ax.contourf(X, Y, truth, cmap = plt.cm.inferno,vmin=0,vmax=100,levels=np.linspace(0, 100, 50))\n",
    "cbar = plt.colorbar(c1);\n",
    "cbar.set_label('Loss Function',rotation = 270,labelpad = 20)\n",
    "plt.xlabel(r'$\\phi_1$'); plt.ylabel(r'$\\phi_2$'); plt.title('Gradient Descent')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Set Parameters for this Model Run and Run the Animation\n",
    "\n",
    "You can change the following parameters and then watch the animation of the gradient descent path.\n",
    "\n",
    "* **r** - learning rate (distance units of the \\[0,100\\] parmaeter space \n",
    "* **momentum - momentum weight (weight on the previous step), from \\[0.0 - no mommentum, 1.0 - next step is previous step\\]\n",
    "* **make_video** - False, run and visualize, True, only make the specified mpeg video file\n",
    "* **x\\[0\\]**, **y\\[0\\]** - initial location for descent path \n",
    "* **delta** - offset applied for numerical differentiation to calculate the local gradient by sampling"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "scrolled": false
   },
   "outputs": [],
   "source": [
    "r = 3.5\n",
    "momentum = 0.9\n",
    "make_video = False\n",
    "x[0] = 25.0; y[0] = 0.0\n",
    "delta = 0.001\n",
    "\n",
    "# set up the plot\n",
    "fig = plt.figure(figsize = (12,10))\n",
    "ax = plt.gca()\n",
    "ax.set_xlim([xmin,xmax]); ax.set_ylim([ymin,ymax])\n",
    "ax.set_xlabel(r'$\\phi_1$'); ax.set_ylabel(r'$\\phi_2$'); ax.set_title('Gradient Descent')\n",
    "plt.rc('xtick', labelsize=10); plt.rc('ytick', labelsize=10) \n",
    "plt.rc('axes', titlesize=20); plt.rc('axes', labelsize=15) \n",
    "point, = ax.plot([],[], marker=\"o\",c=plt.cm.inferno((interp((x[0],y[0])).item())/100), \n",
    "            markeredgecolor = \"white\", ms=15, alpha = 1.0)\n",
    "cbar = plt.colorbar(c1);\n",
    "cbar.set_label('Loss Function',rotation = 270,labelpad = 20)\n",
    "global line\n",
    "\n",
    "x = np.clip(x,xmin+delta,xmax-delta); y = np.clip(y,ymin+delta,ymax-delta) # ensure start is not too close to edge\n",
    "for i in range(1,100):    \n",
    "    gradient_x = (interp((y[i-1], x[i-1]+delta)).item()-interp((y[i-1], x[i-1]-delta)).item())/(2*delta)\n",
    "    gradient_y = (interp((y[i-1]+delta, x[i-1])).item()-interp((y[i-1]-delta, x[i-1])).item())/(2*delta)\n",
    "    \n",
    "    if i > 1: # include mommentum\n",
    "        x_step = momentum*prev_x + (1.0-momentum)*(-1*r*gradient_x) # integrat momentum\n",
    "        y_step = momentum*prev_y + (1.0-momentum)*(-1*r*gradient_y) # integrat momentum\n",
    "    else: \n",
    "        x_step = -1*r*gradient_x \n",
    "        y_step = -1*r*gradient_y       \n",
    "      \n",
    "    x[i] = x[i-1] + x_step                               # step forward\n",
    "    y[i] = y[i-1] + y_step \n",
    "    \n",
    "    x[i] = np.clip(x[i],xmin+delta,xmax-delta); y[i] = np.clip(y[i],ymin+delta,ymax-delta) # ensure point is not too close to edge\n",
    "\n",
    "    prev_x = (x[i] - x[i-1])                              # store current step for momentum on next step\n",
    "    prev_y = (y[i] - y[i-1])\n",
    "               \n",
    "def init():                                             # animation initialization function\n",
    "    ax = plt.gca()\n",
    "#     for c in plt.collections: # possibly better to use: for c in plt.lines (see comment)\n",
    "#         if c.get_gid() == 'line':\n",
    "#             c.remove()\n",
    "    #print(line)\n",
    "    #ax.lines.remove(ax.lines)\n",
    "    c1 = ax.contourf(X, Y, truth, cmap = plt.cm.inferno,vmin=0,vmax=100,levels=np.linspace(0, 100, 50))\n",
    "    ax.contour(X, Y, truth, levels=10, linewidths=0.5, colors='black',linestyles='dashed') \n",
    "#    point, = ax.plot([],[], marker=\"o\",c=plt.cm.inferno((interp((x[0],y[0])).item())/100), markeredgecolor = \"white\", ms=15, alpha = 1.0, label = 'line')\n",
    "    #line.remove()\n",
    "    \n",
    "def update(frame):\n",
    "    N = 5\n",
    "    ax = plt.gca()     \n",
    "    point.set_data(x[frame],y[frame])\n",
    "    point.set_color(plt.cm.inferno((interp((y[frame],x[frame])).item())/100))\n",
    "    line, = plt.plot(x[:frame],y[:frame],c='white',alpha=0.05,linewidth=5)\n",
    "\n",
    "ani = FuncAnimation(fig, update, frames = np.arange(1, len(x)), init_func = init, # make the animation\n",
    "    interval=100, blit=True, repeat = True)\n",
    "\n",
    "if make_video:                                               # set up formatting for the movie files\n",
    "    writer = animation.FFMpegWriter(fps=15, metadata=dict(artist='Me'), bitrate=1800)\n",
    "    ani.save('gradient_descent.mp4', writer=writer)\n",
    "    "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "-5:i"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Interactive Gradient Descent Demo\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "from matplotlib import animation, rc\n",
    "from IPython.display import HTML\n",
    "\n",
    "#rc('animation', html='jshtml')\n",
    "\n",
    "l = widgets.Text(value='                                       Gradient Descent Demo, Prof. Michael Pyrcz, The University of Texas at Austin',\n",
    "                 layout=Layout(width='950px', height='30px'))\n",
    "\n",
    "momentum = widgets.FloatSlider(min=0.0, max = 1.0, value=0.5, step = 0.1, description = 'Momentum',orientation='horizontal',style = {'description_width': 'initial'}, continuous_update=False)\n",
    "r = widgets.FloatSlider(min=1.0, max = 5.0, value=1.5, step = 0.25, description = 'Noise StDev',orientation='horizontal',style = {'description_width': 'initial'}, continuous_update=False)\n",
    "\n",
    "ui = widgets.HBox([momentum,r],)\n",
    "ui2 = widgets.VBox([l,ui],)\n",
    "\n",
    "def run_plot(momentum,r):\n",
    "    make_video = False\n",
    "    nx = 100; ny = 100                                        # size of the loss function grid\n",
    "    cmap = plt.cm.inferno                                     # set the colormap\n",
    "    xmin = 0.0; xmax = 100.0; ymin = 0.0; ymax = 100.0        # set the extents\n",
    "    X, Y = np.mgrid[ymin:ymax:complex(ny), 0:100:complex(nx)]\n",
    "    XY = np.vstack((Y.flatten(), X.flatten())).T              # make location list for interpolation method\n",
    "    XY_list = list(map(tuple, XY))\n",
    "    x = np.zeros(100); y = np.zeros(100)     \n",
    "        \n",
    "    x[0] = 25.0; y[0] = 0.0\n",
    "    delta = 0.001\n",
    "    \n",
    "    # set up the plot\n",
    "    fig = plt.figure(figsize = (12,10))\n",
    "    ax = plt.gca()\n",
    "    ax.set_xlim([xmin,xmax]); ax.set_ylim([ymin,ymax])\n",
    "    ax.set_xlabel(r'$\\phi_1$'); ax.set_ylabel(r'$\\phi_2$'); ax.set_title('Gradient Descent')\n",
    "    plt.rc('xtick', labelsize=10); plt.rc('ytick', labelsize=10) \n",
    "    plt.rc('axes', titlesize=20); plt.rc('axes', labelsize=15) \n",
    "    cbar = plt.colorbar(c1);\n",
    "    cbar.set_label('Loss Function',rotation = 270,labelpad = 20)\n",
    "    point, = ax.plot([],[], marker=\"o\",c=plt.cm.inferno((interp((x[0],y[0])).item())/100), markeredgecolor = \"white\", ms=15, alpha = 1.0)\n",
    "    \n",
    "    x = np.clip(x,xmin+delta,xmax-delta); y = np.clip(y,ymin+delta,ymax-delta) # ensure start is not too close to edge\n",
    "    for i in range(1,100):    \n",
    "        gradient_x = (interp((y[i-1], x[i-1]+delta)).item()-interp((y[i-1], x[i-1]-delta)).item())/(2*delta)\n",
    "        gradient_y = (interp((y[i-1]+delta, x[i-1])).item()-interp((y[i-1]-delta, x[i-1])).item())/(2*delta)\n",
    "        \n",
    "        if i > 1: # include mommentum\n",
    "            x_step = momentum*prev_x + (1.0-momentum)*(-1*r*gradient_x) # integrat momentum\n",
    "            y_step = momentum*prev_y + (1.0-momentum)*(-1*r*gradient_y) # integrat momentum\n",
    "        else: \n",
    "            x_step = -1*r*gradient_x \n",
    "            y_step = -1*r*gradient_y       \n",
    "          \n",
    "        x[i] = x[i-1] + x_step                               # step forward\n",
    "        y[i] = y[i-1] + y_step \n",
    "        \n",
    "        x[i] = np.clip(x[i],xmin+delta,xmax-delta); y[i] = np.clip(y[i],ymin+delta,ymax-delta) # ensure point is not too close to edge\n",
    "    \n",
    "        prev_x = (x[i] - x[i-1])                              # store current step for momentum on next step\n",
    "        prev_y = (y[i] - y[i-1])\n",
    "                   \n",
    "    def init():                                             # animation initialization function\n",
    "        ax = plt.gca()\n",
    "        c1 = ax.contourf(X, Y, truth, cmap = plt.cm.inferno,vmin=0,vmax=100,levels=np.linspace(0, 100, 50))\n",
    "        ax.contour(X, Y, truth, levels=10, linewidths=0.5, colors='black',linestyles='dashed') \n",
    "        \n",
    "    def update(i):\n",
    "        ax = plt.gca()     \n",
    "        point.set_data(x[i],y[i])\n",
    "        point.set_color(plt.cm.inferno((interp((y[i],x[i])).item())/100))\n",
    "        ax.plot(x[:i],y[:i],c='white',alpha=0.05,linewidth=5)\n",
    "    \n",
    "    ani = FuncAnimation(fig, update, np.arange(1, len(x)), init_func = init, # make the animation\n",
    "        interval=10, blit=True, repeat = True)\n",
    "    \n",
    "    #HTML(ani.to_html5_video())\n",
    "    #HTML(ani.to_jshtml())\n",
    "    \n",
    "    if make_video:                                               # set up formatting for the movie files\n",
    "        writer = animation.FFMpegWriter(fps=15, metadata=dict(artist='Me'), bitrate=1800)\n",
    "        ani.save('gradient_descent.mp4', writer=writer)\n",
    "          \n",
    "# connect the function to make the samples and plot to the widgets    \n",
    "interactive_plot = widgets.interactive_output(run_plot, {'momentum':momentum,'r':r})\n",
    "#interactive_plot.clear_output(wait = False)               # reduce flickering by delaying plot updating"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Gradient Descent Optimization for Machine Learning Interactive Demonstation \n",
    "\n",
    "#### Michael Pyrcz, Associate Professo, University of Texas at Austin \n",
    "\n",
    "TBD\n",
    "\n",
    "### The Inputs\n",
    "\n",
    "* **r** - learning rate (distance units of the \\[0,100\\] parmaeter space \n",
    "* **momentum - momentum weight (weight on the previous step), from \\[0.0 - no mommentum, 1.0 - next step is previous step\\]\n",
    "* **make_video** - False, run and visualize, True, only make the specified mpeg video file\n",
    "* **x\\[0\\]**, **y\\[0\\]** - initial location for descent path \n",
    "* **delta** - offset applied for numerical differentiation to calculate the local gradient by sampling\n",
    "\n",
    "Currently this will only make videos, I can't get the animization to show within ipywidgets"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "display(ui2, interactive_plot)                           # display the interactive plot"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Comments\n",
    "\n",
    "This was a basic demonstration of machine learning model training and tuning, model generalization and complexity. I have many other demonstrations and even basics of working with DataFrames, ndarrays, univariate statistics, plotting data, declustering, data transformations and many other workflows available at https://github.com/GeostatsGuy/PythonNumericalDemos and https://github.com/GeostatsGuy/GeostatsPy. \n",
    "  \n",
    "#### The Author:\n",
    "\n",
    "### Michael Pyrcz, Associate Professor, University of Texas at Austin \n",
    "*Novel Data Analytics, Geostatistics and Machine Learning Subsurface Solutions*\n",
    "\n",
    "With over 17 years of experience in subsurface consulting, research and development, Michael has returned to academia driven by his passion for teaching and enthusiasm for enhancing engineers' and geoscientists' impact in subsurface resource development. \n",
    "\n",
    "For more about Michael check out these links:\n",
    "\n",
    "#### [Twitter](https://twitter.com/geostatsguy) | [GitHub](https://github.com/GeostatsGuy) | [Website](http://michaelpyrcz.com) | [GoogleScholar](https://scholar.google.com/citations?user=QVZ20eQAAAAJ&hl=en&oi=ao) | [Book](https://www.amazon.com/Geostatistical-Reservoir-Modeling-Michael-Pyrcz/dp/0199731446) | [YouTube](https://www.youtube.com/channel/UCLqEr-xV-ceHdXXXrTId5ig)  | [LinkedIn](https://www.linkedin.com/in/michael-pyrcz-61a648a1)\n",
    "\n",
    "#### Want to Work Together?\n",
    "\n",
    "I hope this content is helpful to those that want to learn more about subsurface modeling, data analytics and machine learning. Students and working professionals are welcome to participate.\n",
    "\n",
    "* Want to invite me to visit your company for training, mentoring, project review, workflow design and / or consulting? I'd be happy to drop by and work with you! \n",
    "\n",
    "* Interested in partnering, supporting my graduate student research or my Subsurface Data Analytics and Machine Learning consortium (co-PIs including Profs. Foster, Torres-Verdin and van Oort)? My research combines data analytics, stochastic modeling and machine learning theory with practice to develop novel methods and workflows to add value. We are solving challenging subsurface problems!\n",
    "\n",
    "* I can be reached at mpyrcz@austin.utexas.edu.\n",
    "\n",
    "I'm always happy to discuss,\n",
    "\n",
    "*Michael*\n",
    "\n",
    "Michael Pyrcz, Ph.D., P.Eng. Associate Professor The Hildebrand Department of Petroleum and Geosystems Engineering, Bureau of Economic Geology, The Jackson School of Geosciences, The University of Texas at Austin\n",
    "\n",
    "#### More Resources Available at: [Twitter](https://twitter.com/geostatsguy) | [GitHub](https://github.com/GeostatsGuy) | [Website](http://michaelpyrcz.com) | [GoogleScholar](https://scholar.google.com/citations?user=QVZ20eQAAAAJ&hl=en&oi=ao) | [Book](https://www.amazon.com/Geostatistical-Reservoir-Modeling-Michael-Pyrcz/dp/0199731446) | [YouTube](https://www.youtube.com/channel/UCLqEr-xV-ceHdXXXrTId5ig)  | [LinkedIn](https://www.linkedin.com/in/michael-pyrcz-61a648a1)  \n",
    "  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "window.mpl = {};\n",
       "\n",
       "\n",
       "mpl.get_websocket_type = function() {\n",
       "    if (typeof(WebSocket) !== 'undefined') {\n",
       "        return WebSocket;\n",
       "    } else if (typeof(MozWebSocket) !== 'undefined') {\n",
       "        return MozWebSocket;\n",
       "    } else {\n",
       "        alert('Your browser does not have WebSocket support. ' +\n",
       "              'Please try Chrome, Safari or Firefox ≥ 6. ' +\n",
       "              'Firefox 4 and 5 are also supported but you ' +\n",
       "              'have to enable WebSockets in about:config.');\n",
       "    };\n",
       "}\n",
       "\n",
       "mpl.figure = function(figure_id, websocket, ondownload, parent_element) {\n",
       "    this.id = figure_id;\n",
       "\n",
       "    this.ws = websocket;\n",
       "\n",
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       "        var warnings = document.getElementById(\"mpl-warnings\");\n",
       "        if (warnings) {\n",
       "            warnings.style.display = 'block';\n",
       "            warnings.textContent = (\n",
       "                \"This browser does not support binary websocket messages. \" +\n",
       "                    \"Performance may be slow.\");\n",
       "        }\n",
       "    }\n",
       "\n",
       "    this.imageObj = new Image();\n",
       "\n",
       "    this.context = undefined;\n",
       "    this.message = undefined;\n",
       "    this.canvas = undefined;\n",
       "    this.rubberband_canvas = undefined;\n",
       "    this.rubberband_context = undefined;\n",
       "    this.format_dropdown = undefined;\n",
       "\n",
       "    this.image_mode = 'full';\n",
       "\n",
       "    this.root = $('<div/>');\n",
       "    this._root_extra_style(this.root)\n",
       "    this.root.attr('style', 'display: inline-block');\n",
       "\n",
       "    $(parent_element).append(this.root);\n",
       "\n",
       "    this._init_header(this);\n",
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       "    this._init_toolbar(this);\n",
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       "    var fig = this;\n",
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       "    this.waiting = false;\n",
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       "    this.ws.onopen =  function () {\n",
       "            fig.send_message(\"supports_binary\", {value: fig.supports_binary});\n",
       "            fig.send_message(\"send_image_mode\", {});\n",
       "            if (mpl.ratio != 1) {\n",
       "                fig.send_message(\"set_dpi_ratio\", {'dpi_ratio': mpl.ratio});\n",
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       "            fig.send_message(\"refresh\", {});\n",
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       "\n",
       "mpl.figure.prototype._init_header = function() {\n",
       "    var titlebar = $(\n",
       "        '<div class=\"ui-dialog-titlebar ui-widget-header ui-corner-all ' +\n",
       "        'ui-helper-clearfix\"/>');\n",
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       "        '<div class=\"ui-dialog-title\" style=\"width: 100%; ' +\n",
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       "        return fig.key_event(event, event['data']);\n",
       "    }\n",
       "\n",
       "    canvas_div.keydown('key_press', canvas_keyboard_event);\n",
       "    canvas_div.keyup('key_release', canvas_keyboard_event);\n",
       "    this.canvas_div = canvas_div\n",
       "    this._canvas_extra_style(canvas_div)\n",
       "    this.root.append(canvas_div);\n",
       "\n",
       "    var canvas = $('<canvas/>');\n",
       "    canvas.addClass('mpl-canvas');\n",
       "    canvas.attr('style', \"left: 0; top: 0; z-index: 0; outline: 0\")\n",
       "\n",
       "    this.canvas = canvas[0];\n",
       "    this.context = canvas[0].getContext(\"2d\");\n",
       "\n",
       "    var backingStore = this.context.backingStorePixelRatio ||\n",
       "\tthis.context.webkitBackingStorePixelRatio ||\n",
       "\tthis.context.mozBackingStorePixelRatio ||\n",
       "\tthis.context.msBackingStorePixelRatio ||\n",
       "\tthis.context.oBackingStorePixelRatio ||\n",
       "\tthis.context.backingStorePixelRatio || 1;\n",
       "\n",
       "    mpl.ratio = (window.devicePixelRatio || 1) / backingStore;\n",
       "\n",
       "    var rubberband = $('<canvas/>');\n",
       "    rubberband.attr('style', \"position: absolute; left: 0; top: 0; z-index: 1;\")\n",
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       "\n",
       "    canvas_div.resizable({\n",
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       "        resize: function(event, ui) {\n",
       "            fig.request_resize(ui.size.width, ui.size.height);\n",
       "        },\n",
       "        stop: function(event, ui) {\n",
       "            pass_mouse_events = true;\n",
       "            fig.request_resize(ui.size.width, ui.size.height);\n",
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       "\n",
       "    function mouse_event_fn(event) {\n",
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       "\n",
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       "    rubberband.mouseup('button_release', mouse_event_fn);\n",
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       "\n",
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       "\n",
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       "    this.rubberband_context.strokeStyle = \"#000000\";\n",
       "\n",
       "    this._resize_canvas = function(width, height) {\n",
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       "        // canvas in synch.\n",
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       "\n",
       "        canvas.attr('width', width * mpl.ratio);\n",
       "        canvas.attr('height', height * mpl.ratio);\n",
       "        canvas.attr('style', 'width: ' + width + 'px; height: ' + height + 'px;');\n",
       "\n",
       "        rubberband.attr('width', width);\n",
       "        rubberband.attr('height', height);\n",
       "    }\n",
       "\n",
       "    // Set the figure to an initial 600x600px, this will subsequently be updated\n",
       "    // upon first draw.\n",
       "    this._resize_canvas(600, 600);\n",
       "\n",
       "    // Disable right mouse context menu.\n",
       "    $(this.rubberband_canvas).bind(\"contextmenu\",function(e){\n",
       "        return false;\n",
       "    });\n",
       "\n",
       "    function set_focus () {\n",
       "        canvas.focus();\n",
       "        canvas_div.focus();\n",
       "    }\n",
       "\n",
       "    window.setTimeout(set_focus, 100);\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._init_toolbar = function() {\n",
       "    var fig = this;\n",
       "\n",
       "    var nav_element = $('<div/>');\n",
       "    nav_element.attr('style', 'width: 100%');\n",
       "    this.root.append(nav_element);\n",
       "\n",
       "    // Define a callback function for later on.\n",
       "    function toolbar_event(event) {\n",
       "        return fig.toolbar_button_onclick(event['data']);\n",
       "    }\n",
       "    function toolbar_mouse_event(event) {\n",
       "        return fig.toolbar_button_onmouseover(event['data']);\n",
       "    }\n",
       "\n",
       "    for(var toolbar_ind in mpl.toolbar_items) {\n",
       "        var name = mpl.toolbar_items[toolbar_ind][0];\n",
       "        var tooltip = mpl.toolbar_items[toolbar_ind][1];\n",
       "        var image = mpl.toolbar_items[toolbar_ind][2];\n",
       "        var method_name = mpl.toolbar_items[toolbar_ind][3];\n",
       "\n",
       "        if (!name) {\n",
       "            // put a spacer in here.\n",
       "            continue;\n",
       "        }\n",
       "        var button = $('<button/>');\n",
       "        button.addClass('ui-button ui-widget ui-state-default ui-corner-all ' +\n",
       "                        'ui-button-icon-only');\n",
       "        button.attr('role', 'button');\n",
       "        button.attr('aria-disabled', 'false');\n",
       "        button.click(method_name, toolbar_event);\n",
       "        button.mouseover(tooltip, toolbar_mouse_event);\n",
       "\n",
       "        var icon_img = $('<span/>');\n",
       "        icon_img.addClass('ui-button-icon-primary ui-icon');\n",
       "        icon_img.addClass(image);\n",
       "        icon_img.addClass('ui-corner-all');\n",
       "\n",
       "        var tooltip_span = $('<span/>');\n",
       "        tooltip_span.addClass('ui-button-text');\n",
       "        tooltip_span.html(tooltip);\n",
       "\n",
       "        button.append(icon_img);\n",
       "        button.append(tooltip_span);\n",
       "\n",
       "        nav_element.append(button);\n",
       "    }\n",
       "\n",
       "    var fmt_picker_span = $('<span/>');\n",
       "\n",
       "    var fmt_picker = $('<select/>');\n",
       "    fmt_picker.addClass('mpl-toolbar-option ui-widget ui-widget-content');\n",
       "    fmt_picker_span.append(fmt_picker);\n",
       "    nav_element.append(fmt_picker_span);\n",
       "    this.format_dropdown = fmt_picker[0];\n",
       "\n",
       "    for (var ind in mpl.extensions) {\n",
       "        var fmt = mpl.extensions[ind];\n",
       "        var option = $(\n",
       "            '<option/>', {selected: fmt === mpl.default_extension}).html(fmt);\n",
       "        fmt_picker.append(option);\n",
       "    }\n",
       "\n",
       "    // Add hover states to the ui-buttons\n",
       "    $( \".ui-button\" ).hover(\n",
       "        function() { $(this).addClass(\"ui-state-hover\");},\n",
       "        function() { $(this).removeClass(\"ui-state-hover\");}\n",
       "    );\n",
       "\n",
       "    var status_bar = $('<span class=\"mpl-message\"/>');\n",
       "    nav_element.append(status_bar);\n",
       "    this.message = status_bar[0];\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.request_resize = function(x_pixels, y_pixels) {\n",
       "    // Request matplotlib to resize the figure. Matplotlib will then trigger a resize in the client,\n",
       "    // which will in turn request a refresh of the image.\n",
       "    this.send_message('resize', {'width': x_pixels, 'height': y_pixels});\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.send_message = function(type, properties) {\n",
       "    properties['type'] = type;\n",
       "    properties['figure_id'] = this.id;\n",
       "    this.ws.send(JSON.stringify(properties));\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.send_draw_message = function() {\n",
       "    if (!this.waiting) {\n",
       "        this.waiting = true;\n",
       "        this.ws.send(JSON.stringify({type: \"draw\", figure_id: this.id}));\n",
       "    }\n",
       "}\n",
       "\n",
       "\n",
       "mpl.figure.prototype.handle_save = function(fig, msg) {\n",
       "    var format_dropdown = fig.format_dropdown;\n",
       "    var format = format_dropdown.options[format_dropdown.selectedIndex].value;\n",
       "    fig.ondownload(fig, format);\n",
       "}\n",
       "\n",
       "\n",
       "mpl.figure.prototype.handle_resize = function(fig, msg) {\n",
       "    var size = msg['size'];\n",
       "    if (size[0] != fig.canvas.width || size[1] != fig.canvas.height) {\n",
       "        fig._resize_canvas(size[0], size[1]);\n",
       "        fig.send_message(\"refresh\", {});\n",
       "    };\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_rubberband = function(fig, msg) {\n",
       "    var x0 = msg['x0'] / mpl.ratio;\n",
       "    var y0 = (fig.canvas.height - msg['y0']) / mpl.ratio;\n",
       "    var x1 = msg['x1'] / mpl.ratio;\n",
       "    var y1 = (fig.canvas.height - msg['y1']) / mpl.ratio;\n",
       "    x0 = Math.floor(x0) + 0.5;\n",
       "    y0 = Math.floor(y0) + 0.5;\n",
       "    x1 = Math.floor(x1) + 0.5;\n",
       "    y1 = Math.floor(y1) + 0.5;\n",
       "    var min_x = Math.min(x0, x1);\n",
       "    var min_y = Math.min(y0, y1);\n",
       "    var width = Math.abs(x1 - x0);\n",
       "    var height = Math.abs(y1 - y0);\n",
       "\n",
       "    fig.rubberband_context.clearRect(\n",
       "        0, 0, fig.canvas.width / mpl.ratio, fig.canvas.height / mpl.ratio);\n",
       "\n",
       "    fig.rubberband_context.strokeRect(min_x, min_y, width, height);\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_figure_label = function(fig, msg) {\n",
       "    // Updates the figure title.\n",
       "    fig.header.textContent = msg['label'];\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_cursor = function(fig, msg) {\n",
       "    var cursor = msg['cursor'];\n",
       "    switch(cursor)\n",
       "    {\n",
       "    case 0:\n",
       "        cursor = 'pointer';\n",
       "        break;\n",
       "    case 1:\n",
       "        cursor = 'default';\n",
       "        break;\n",
       "    case 2:\n",
       "        cursor = 'crosshair';\n",
       "        break;\n",
       "    case 3:\n",
       "        cursor = 'move';\n",
       "        break;\n",
       "    }\n",
       "    fig.rubberband_canvas.style.cursor = cursor;\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_message = function(fig, msg) {\n",
       "    fig.message.textContent = msg['message'];\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_draw = function(fig, msg) {\n",
       "    // Request the server to send over a new figure.\n",
       "    fig.send_draw_message();\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_image_mode = function(fig, msg) {\n",
       "    fig.image_mode = msg['mode'];\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.updated_canvas_event = function() {\n",
       "    // Called whenever the canvas gets updated.\n",
       "    this.send_message(\"ack\", {});\n",
       "}\n",
       "\n",
       "// A function to construct a web socket function for onmessage handling.\n",
       "// Called in the figure constructor.\n",
       "mpl.figure.prototype._make_on_message_function = function(fig) {\n",
       "    return function socket_on_message(evt) {\n",
       "        if (evt.data instanceof Blob) {\n",
       "            /* FIXME: We get \"Resource interpreted as Image but\n",
       "             * transferred with MIME type text/plain:\" errors on\n",
       "             * Chrome.  But how to set the MIME type?  It doesn't seem\n",
       "             * to be part of the websocket stream */\n",
       "            evt.data.type = \"image/png\";\n",
       "\n",
       "            /* Free the memory for the previous frames */\n",
       "            if (fig.imageObj.src) {\n",
       "                (window.URL || window.webkitURL).revokeObjectURL(\n",
       "                    fig.imageObj.src);\n",
       "            }\n",
       "\n",
       "            fig.imageObj.src = (window.URL || window.webkitURL).createObjectURL(\n",
       "                evt.data);\n",
       "            fig.updated_canvas_event();\n",
       "            fig.waiting = false;\n",
       "            return;\n",
       "        }\n",
       "        else if (typeof evt.data === 'string' && evt.data.slice(0, 21) == \"data:image/png;base64\") {\n",
       "            fig.imageObj.src = evt.data;\n",
       "            fig.updated_canvas_event();\n",
       "            fig.waiting = false;\n",
       "            return;\n",
       "        }\n",
       "\n",
       "        var msg = JSON.parse(evt.data);\n",
       "        var msg_type = msg['type'];\n",
       "\n",
       "        // Call the  \"handle_{type}\" callback, which takes\n",
       "        // the figure and JSON message as its only arguments.\n",
       "        try {\n",
       "            var callback = fig[\"handle_\" + msg_type];\n",
       "        } catch (e) {\n",
       "            console.log(\"No handler for the '\" + msg_type + \"' message type: \", msg);\n",
       "            return;\n",
       "        }\n",
       "\n",
       "        if (callback) {\n",
       "            try {\n",
       "                // console.log(\"Handling '\" + msg_type + \"' message: \", msg);\n",
       "                callback(fig, msg);\n",
       "            } catch (e) {\n",
       "                console.log(\"Exception inside the 'handler_\" + msg_type + \"' callback:\", e, e.stack, msg);\n",
       "            }\n",
       "        }\n",
       "    };\n",
       "}\n",
       "\n",
       "// from http://stackoverflow.com/questions/1114465/getting-mouse-location-in-canvas\n",
       "mpl.findpos = function(e) {\n",
       "    //this section is from http://www.quirksmode.org/js/events_properties.html\n",
       "    var targ;\n",
       "    if (!e)\n",
       "        e = window.event;\n",
       "    if (e.target)\n",
       "        targ = e.target;\n",
       "    else if (e.srcElement)\n",
       "        targ = e.srcElement;\n",
       "    if (targ.nodeType == 3) // defeat Safari bug\n",
       "        targ = targ.parentNode;\n",
       "\n",
       "    // jQuery normalizes the pageX and pageY\n",
       "    // pageX,Y are the mouse positions relative to the document\n",
       "    // offset() returns the position of the element relative to the document\n",
       "    var x = e.pageX - $(targ).offset().left;\n",
       "    var y = e.pageY - $(targ).offset().top;\n",
       "\n",
       "    return {\"x\": x, \"y\": y};\n",
       "};\n",
       "\n",
       "/*\n",
       " * return a copy of an object with only non-object keys\n",
       " * we need this to avoid circular references\n",
       " * http://stackoverflow.com/a/24161582/3208463\n",
       " */\n",
       "function simpleKeys (original) {\n",
       "  return Object.keys(original).reduce(function (obj, key) {\n",
       "    if (typeof original[key] !== 'object')\n",
       "        obj[key] = original[key]\n",
       "    return obj;\n",
       "  }, {});\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.mouse_event = function(event, name) {\n",
       "    var canvas_pos = mpl.findpos(event)\n",
       "\n",
       "    if (name === 'button_press')\n",
       "    {\n",
       "        this.canvas.focus();\n",
       "        this.canvas_div.focus();\n",
       "    }\n",
       "\n",
       "    var x = canvas_pos.x * mpl.ratio;\n",
       "    var y = canvas_pos.y * mpl.ratio;\n",
       "\n",
       "    this.send_message(name, {x: x, y: y, button: event.button,\n",
       "                             step: event.step,\n",
       "                             guiEvent: simpleKeys(event)});\n",
       "\n",
       "    /* This prevents the web browser from automatically changing to\n",
       "     * the text insertion cursor when the button is pressed.  We want\n",
       "     * to control all of the cursor setting manually through the\n",
       "     * 'cursor' event from matplotlib */\n",
       "    event.preventDefault();\n",
       "    return false;\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._key_event_extra = function(event, name) {\n",
       "    // Handle any extra behaviour associated with a key event\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.key_event = function(event, name) {\n",
       "\n",
       "    // Prevent repeat events\n",
       "    if (name == 'key_press')\n",
       "    {\n",
       "        if (event.which === this._key)\n",
       "            return;\n",
       "        else\n",
       "            this._key = event.which;\n",
       "    }\n",
       "    if (name == 'key_release')\n",
       "        this._key = null;\n",
       "\n",
       "    var value = '';\n",
       "    if (event.ctrlKey && event.which != 17)\n",
       "        value += \"ctrl+\";\n",
       "    if (event.altKey && event.which != 18)\n",
       "        value += \"alt+\";\n",
       "    if (event.shiftKey && event.which != 16)\n",
       "        value += \"shift+\";\n",
       "\n",
       "    value += 'k';\n",
       "    value += event.which.toString();\n",
       "\n",
       "    this._key_event_extra(event, name);\n",
       "\n",
       "    this.send_message(name, {key: value,\n",
       "                             guiEvent: simpleKeys(event)});\n",
       "    return false;\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.toolbar_button_onclick = function(name) {\n",
       "    if (name == 'download') {\n",
       "        this.handle_save(this, null);\n",
       "    } else {\n",
       "        this.send_message(\"toolbar_button\", {name: name});\n",
       "    }\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.toolbar_button_onmouseover = function(tooltip) {\n",
       "    this.message.textContent = tooltip;\n",
       "};\n",
       "mpl.toolbar_items = [[\"Home\", \"Reset original view\", \"fa fa-home icon-home\", \"home\"], [\"Back\", \"Back to previous view\", \"fa fa-arrow-left icon-arrow-left\", \"back\"], [\"Forward\", \"Forward to next view\", \"fa fa-arrow-right icon-arrow-right\", \"forward\"], [\"\", \"\", \"\", \"\"], [\"Pan\", \"Pan axes with left mouse, zoom with right\", \"fa fa-arrows icon-move\", \"pan\"], [\"Zoom\", \"Zoom to rectangle\", \"fa fa-square-o icon-check-empty\", \"zoom\"], [\"\", \"\", \"\", \"\"], [\"Download\", \"Download plot\", \"fa fa-floppy-o icon-save\", \"download\"]];\n",
       "\n",
       "mpl.extensions = [\"eps\", \"jpeg\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\", \"tif\"];\n",
       "\n",
       "mpl.default_extension = \"png\";var comm_websocket_adapter = function(comm) {\n",
       "    // Create a \"websocket\"-like object which calls the given IPython comm\n",
       "    // object with the appropriate methods. Currently this is a non binary\n",
       "    // socket, so there is still some room for performance tuning.\n",
       "    var ws = {};\n",
       "\n",
       "    ws.close = function() {\n",
       "        comm.close()\n",
       "    };\n",
       "    ws.send = function(m) {\n",
       "        //console.log('sending', m);\n",
       "        comm.send(m);\n",
       "    };\n",
       "    // Register the callback with on_msg.\n",
       "    comm.on_msg(function(msg) {\n",
       "        //console.log('receiving', msg['content']['data'], msg);\n",
       "        // Pass the mpl event to the overridden (by mpl) onmessage function.\n",
       "        ws.onmessage(msg['content']['data'])\n",
       "    });\n",
       "    return ws;\n",
       "}\n",
       "\n",
       "mpl.mpl_figure_comm = function(comm, msg) {\n",
       "    // This is the function which gets called when the mpl process\n",
       "    // starts-up an IPython Comm through the \"matplotlib\" channel.\n",
       "\n",
       "    var id = msg.content.data.id;\n",
       "    // Get hold of the div created by the display call when the Comm\n",
       "    // socket was opened in Python.\n",
       "    var element = $(\"#\" + id);\n",
       "    var ws_proxy = comm_websocket_adapter(comm)\n",
       "\n",
       "    function ondownload(figure, format) {\n",
       "        window.open(figure.imageObj.src);\n",
       "    }\n",
       "\n",
       "    var fig = new mpl.figure(id, ws_proxy,\n",
       "                           ondownload,\n",
       "                           element.get(0));\n",
       "\n",
       "    // Call onopen now - mpl needs it, as it is assuming we've passed it a real\n",
       "    // web socket which is closed, not our websocket->open comm proxy.\n",
       "    ws_proxy.onopen();\n",
       "\n",
       "    fig.parent_element = element.get(0);\n",
       "    fig.cell_info = mpl.find_output_cell(\"<div id='\" + id + \"'></div>\");\n",
       "    if (!fig.cell_info) {\n",
       "        console.error(\"Failed to find cell for figure\", id, fig);\n",
       "        return;\n",
       "    }\n",
       "\n",
       "    var output_index = fig.cell_info[2]\n",
       "    var cell = fig.cell_info[0];\n",
       "\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.handle_close = function(fig, msg) {\n",
       "    var width = fig.canvas.width/mpl.ratio\n",
       "    fig.root.unbind('remove')\n",
       "\n",
       "    // Update the output cell to use the data from the current canvas.\n",
       "    fig.push_to_output();\n",
       "    var dataURL = fig.canvas.toDataURL();\n",
       "    // Re-enable the keyboard manager in IPython - without this line, in FF,\n",
       "    // the notebook keyboard shortcuts fail.\n",
       "    IPython.keyboard_manager.enable()\n",
       "    $(fig.parent_element).html('<img src=\"' + dataURL + '\" width=\"' + width + '\">');\n",
       "    fig.close_ws(fig, msg);\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.close_ws = function(fig, msg){\n",
       "    fig.send_message('closing', msg);\n",
       "    // fig.ws.close()\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.push_to_output = function(remove_interactive) {\n",
       "    // Turn the data on the canvas into data in the output cell.\n",
       "    var width = this.canvas.width/mpl.ratio\n",
       "    var dataURL = this.canvas.toDataURL();\n",
       "    this.cell_info[1]['text/html'] = '<img src=\"' + dataURL + '\" width=\"' + width + '\">';\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.updated_canvas_event = function() {\n",
       "    // Tell IPython that the notebook contents must change.\n",
       "    IPython.notebook.set_dirty(true);\n",
       "    this.send_message(\"ack\", {});\n",
       "    var fig = this;\n",
       "    // Wait a second, then push the new image to the DOM so\n",
       "    // that it is saved nicely (might be nice to debounce this).\n",
       "    setTimeout(function () { fig.push_to_output() }, 1000);\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._init_toolbar = function() {\n",
       "    var fig = this;\n",
       "\n",
       "    var nav_element = $('<div/>');\n",
       "    nav_element.attr('style', 'width: 100%');\n",
       "    this.root.append(nav_element);\n",
       "\n",
       "    // Define a callback function for later on.\n",
       "    function toolbar_event(event) {\n",
       "        return fig.toolbar_button_onclick(event['data']);\n",
       "    }\n",
       "    function toolbar_mouse_event(event) {\n",
       "        return fig.toolbar_button_onmouseover(event['data']);\n",
       "    }\n",
       "\n",
       "    for(var toolbar_ind in mpl.toolbar_items){\n",
       "        var name = mpl.toolbar_items[toolbar_ind][0];\n",
       "        var tooltip = mpl.toolbar_items[toolbar_ind][1];\n",
       "        var image = mpl.toolbar_items[toolbar_ind][2];\n",
       "        var method_name = mpl.toolbar_items[toolbar_ind][3];\n",
       "\n",
       "        if (!name) { continue; };\n",
       "\n",
       "        var button = $('<button class=\"btn btn-default\" href=\"#\" title=\"' + name + '\"><i class=\"fa ' + image + ' fa-lg\"></i></button>');\n",
       "        button.click(method_name, toolbar_event);\n",
       "        button.mouseover(tooltip, toolbar_mouse_event);\n",
       "        nav_element.append(button);\n",
       "    }\n",
       "\n",
       "    // Add the status bar.\n",
       "    var status_bar = $('<span class=\"mpl-message\" style=\"text-align:right; float: right;\"/>');\n",
       "    nav_element.append(status_bar);\n",
       "    this.message = status_bar[0];\n",
       "\n",
       "    // Add the close button to the window.\n",
       "    var buttongrp = $('<div class=\"btn-group inline pull-right\"></div>');\n",
       "    var button = $('<button class=\"btn btn-mini btn-primary\" href=\"#\" title=\"Stop Interaction\"><i class=\"fa fa-power-off icon-remove icon-large\"></i></button>');\n",
       "    button.click(function (evt) { fig.handle_close(fig, {}); } );\n",
       "    button.mouseover('Stop Interaction', toolbar_mouse_event);\n",
       "    buttongrp.append(button);\n",
       "    var titlebar = this.root.find($('.ui-dialog-titlebar'));\n",
       "    titlebar.prepend(buttongrp);\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._root_extra_style = function(el){\n",
       "    var fig = this\n",
       "    el.on(\"remove\", function(){\n",
       "\tfig.close_ws(fig, {});\n",
       "    });\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._canvas_extra_style = function(el){\n",
       "    // this is important to make the div 'focusable\n",
       "    el.attr('tabindex', 0)\n",
       "    // reach out to IPython and tell the keyboard manager to turn it's self\n",
       "    // off when our div gets focus\n",
       "\n",
       "    // location in version 3\n",
       "    if (IPython.notebook.keyboard_manager) {\n",
       "        IPython.notebook.keyboard_manager.register_events(el);\n",
       "    }\n",
       "    else {\n",
       "        // location in version 2\n",
       "        IPython.keyboard_manager.register_events(el);\n",
       "    }\n",
       "\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._key_event_extra = function(event, name) {\n",
       "    var manager = IPython.notebook.keyboard_manager;\n",
       "    if (!manager)\n",
       "        manager = IPython.keyboard_manager;\n",
       "\n",
       "    // Check for shift+enter\n",
       "    if (event.shiftKey && event.which == 13) {\n",
       "        this.canvas_div.blur();\n",
       "        // select the cell after this one\n",
       "        var index = IPython.notebook.find_cell_index(this.cell_info[0]);\n",
       "        IPython.notebook.select(index + 1);\n",
       "    }\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_save = function(fig, msg) {\n",
       "    fig.ondownload(fig, null);\n",
       "}\n",
       "\n",
       "\n",
       "mpl.find_output_cell = function(html_output) {\n",
       "    // Return the cell and output element which can be found *uniquely* in the notebook.\n",
       "    // Note - this is a bit hacky, but it is done because the \"notebook_saving.Notebook\"\n",
       "    // IPython event is triggered only after the cells have been serialised, which for\n",
       "    // our purposes (turning an active figure into a static one), is too late.\n",
       "    var cells = IPython.notebook.get_cells();\n",
       "    var ncells = cells.length;\n",
       "    for (var i=0; i<ncells; i++) {\n",
       "        var cell = cells[i];\n",
       "        if (cell.cell_type === 'code'){\n",
       "            for (var j=0; j<cell.output_area.outputs.length; j++) {\n",
       "                var data = cell.output_area.outputs[j];\n",
       "                if (data.data) {\n",
       "                    // IPython >= 3 moved mimebundle to data attribute of output\n",
       "                    data = data.data;\n",
       "                }\n",
       "                if (data['text/html'] == html_output) {\n",
       "                    return [cell, data, j];\n",
       "                }\n",
       "            }\n",
       "        }\n",
       "    }\n",
       "}\n",
       "\n",
       "// Register the function which deals with the matplotlib target/channel.\n",
       "// The kernel may be null if the page has been refreshed.\n",
       "if (IPython.notebook.kernel != null) {\n",
       "    IPython.notebook.kernel.comm_manager.register_target('matplotlib', mpl.mpl_figure_comm);\n",
       "}\n"
      ],
      "text/plain": [
       "<IPython.core.display.Javascript object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<img src=\"data:image/png;base64,[encoded data, 116720 characters]\" width=\"863.1833333333333\">"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "r0 = 10.0\n",
    "momentum = 0.9\n",
    "make_video = False\n",
    "\n",
    "delta = 0.001\n",
    "k = 0.04                                         # exponential decay parameter\n",
    "\n",
    "x = np.zeros(100); y = np.zeros(100)\n",
    "x[0] = 5.0; y[0] = 50.0\n",
    "\n",
    "plt.close()\n",
    "# set up the plot\n",
    "fig = plt.figure(figsize = (12,10))\n",
    "ax = plt.gca()\n",
    "ax.set_xlim([xmin,xmax]); ax.set_ylim([ymin,ymax])\n",
    "ax.set_xlabel(r'$\\phi_1$'); ax.set_ylabel(r'$\\phi_2$'); ax.set_title('Gradient Descent')\n",
    "plt.rc('xtick', labelsize=10); plt.rc('ytick', labelsize=10) \n",
    "plt.rc('axes', titlesize=20); plt.rc('axes', labelsize=15) \n",
    "point, = ax.plot([],[], marker=\"o\",c='white', \n",
    "            markeredgecolor = \"black\", ms=30, alpha = 0.5)\n",
    "cbar = plt.colorbar(c1);\n",
    "cbar.set_label('Loss Function',rotation = 270,labelpad = 20)\n",
    "global line\n",
    "\n",
    "x = np.clip(x,xmin+delta,xmax-delta); y = np.clip(y,ymin+delta,ymax-delta) # ensure start is not too close to edge\n",
    "for i in range(1,100):    \n",
    "    gradient_x = (interp((y[i-1], x[i-1]+delta)).item()-interp((y[i-1], x[i-1]-delta)).item())/(2*delta)\n",
    "    gradient_y = (interp((y[i-1]+delta, x[i-1])).item()-interp((y[i-1]-delta, x[i-1])).item())/(2*delta)\n",
    "    \n",
    "    r = r0 * math.exp(-k*i)    \n",
    "    \n",
    "    if i > 1: # include mommentum\n",
    "        x_step = momentum*prev_x + (1.0-momentum)*(-1*r*gradient_x) # integrat momentum\n",
    "        y_step = momentum*prev_y + (1.0-momentum)*(-1*r*gradient_y) # integrat momentum\n",
    "    else: \n",
    "        x_step = -1*r*gradient_x \n",
    "        y_step = -1*r*gradient_y \n",
    "      \n",
    "    x[i] = x[i-1] + x_step                               # step forward\n",
    "    y[i] = y[i-1] + y_step \n",
    "    \n",
    "    x[i] = np.clip(x[i],xmin+delta,xmax-delta); y[i] = np.clip(y[i],ymin+delta,ymax-delta) # ensure point is not too close to edge\n",
    "\n",
    "    prev_x = (x[i] - x[i-1])                              # store current step for momentum on next step\n",
    "    prev_y = (y[i] - y[i-1])\n",
    "               \n",
    "def init():                                             # animation initialization function\n",
    "    ax = plt.gca()\n",
    "    c1 = ax.contourf(X, Y, truth, cmap = plt.cm.inferno,vmin=0,vmax=100,levels=np.linspace(0, 100, 50))\n",
    "    ax.contour(X, Y, truth, levels=10, linewidths=0.5, colors='black',linestyles='dashed') \n",
    "    \n",
    "def update(frame):\n",
    "    N = 10\n",
    "    ax = plt.gca()\n",
    "    N = min(N,frame)\n",
    "    top = min(frame,90)\n",
    "    point.set_data(x[frame-N:top],y[frame-N:top])\n",
    "    #point.set_sizes(np.linspace(1,N,N))\n",
    "    #point.set_color(plt.cm.inferno((interp((y[frame],x[frame])).item())/100))\n",
    "\n",
    "ani = FuncAnimation(fig, update, frames = np.arange(1, len(x)-20), init_func = init, # make the animation\n",
    "    interval=100, blit=True, repeat = False)\n",
    "\n",
    "if make_video:                                               # set up formatting for the movie files\n",
    "    writer = animation.FFMpegWriter(fps=15, metadata=dict(artist='Michael Pyrcz'), bitrate=100000)\n",
    "    ani.save('gradient_descent.mp4', writer=writer)\n",
    "    "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([5.00000000e+00, 1.09732151e+01, 1.66005737e+01, 2.26402826e+01,\n",
       "       2.74773594e+01, 3.24681367e+01, 3.65755385e+01, 4.16798377e+01,\n",
       "       4.70823671e+01, 5.08227895e+01, 5.29042218e+01, 5.35516769e+01,\n",
       "       5.29814415e+01, 5.18579690e+01, 5.00922320e+01, 4.83208676e+01,\n",
       "       4.63086883e+01, 4.42477544e+01, 4.25447098e+01, 4.13299349e+01,\n",
       "       4.15426236e+01, 4.23916660e+01, 4.37224425e+01, 4.47674252e+01,\n",
       "       4.53579521e+01, 4.55815336e+01, 4.53184349e+01, 4.48883008e+01,\n",
       "       4.43364513e+01, 4.39017758e+01, 4.35581920e+01, 4.32781110e+01,\n",
       "       4.32553479e+01, 4.34076260e+01, 4.36934037e+01, 4.38853347e+01,\n",
       "       4.40235862e+01, 4.41277874e+01, 4.42048521e+01, 4.42534424e+01,\n",
       "       4.42670355e+01, 4.42565021e+01, 4.42230012e+01, 4.41817463e+01,\n",
       "       4.41360576e+01, 4.40815382e+01, 4.40084635e+01, 4.39253691e+01,\n",
       "       4.38506894e+01, 4.37888653e+01, 1.00000000e-03, 1.00000000e-03,\n",
       "       1.00000000e-03, 1.00000000e-03, 1.00000000e-03, 1.00000000e-03,\n",
       "       1.00000000e-03, 1.00000000e-03, 1.00000000e-03, 1.00000000e-03,\n",
       "       1.00000000e-03, 1.00000000e-03, 1.00000000e-03, 1.00000000e-03,\n",
       "       1.00000000e-03, 1.00000000e-03, 1.00000000e-03, 1.00000000e-03,\n",
       "       1.00000000e-03, 1.00000000e-03, 1.00000000e-03, 1.00000000e-03,\n",
       "       1.00000000e-03, 1.00000000e-03, 1.00000000e-03, 1.00000000e-03,\n",
       "       1.00000000e-03, 1.00000000e-03, 1.00000000e-03, 1.00000000e-03,\n",
       "       1.00000000e-03, 1.00000000e-03, 1.00000000e-03, 1.00000000e-03,\n",
       "       1.00000000e-03, 1.00000000e-03, 1.00000000e-03, 1.00000000e-03,\n",
       "       1.00000000e-03, 1.00000000e-03, 1.00000000e-03, 1.00000000e-03,\n",
       "       1.00000000e-03, 1.00000000e-03, 1.00000000e-03, 1.00000000e-03,\n",
       "       1.00000000e-03, 1.00000000e-03, 1.00000000e-03, 1.00000000e-03])"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x[90-20:80]\n",
    "x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "from matplotlib import animation, rc\n",
    "from IPython.display import HTML\n",
    "\n",
    "rc('animation', html='jshtml')\n",
    "\n",
    "l = widgets.Text(value='                                       Gradient Descent Demo, Prof. Michael Pyrcz, The University of Texas at Austin',\n",
    "                 layout=Layout(width='950px', height='30px'))\n",
    "\n",
    "momentum = widgets.FloatSlider(min=0.0, max = 1.0, value=0.5, step = 0.1, description = 'Momentum',orientation='horizontal',style = {'description_width': 'initial'}, continuous_update=False)\n",
    "r = widgets.FloatSlider(min=1.0, max = 5.0, value=1.5, step = 0.25, description = 'Noise StDev',orientation='horizontal',style = {'description_width': 'initial'}, continuous_update=False)\n",
    "\n",
    "ui = widgets.HBox([momentum,r],)\n",
    "ui2 = widgets.VBox([l,ui],)\n",
    "\n",
    "def run_plot(momentum,r):\n",
    "    make_video = False\n",
    "    nx = 100; ny = 100                                        # size of the loss function grid\n",
    "    cmap = plt.cm.inferno                                     # set the colormap\n",
    "    xmin = 0.0; xmax = 100.0; ymin = 0.0; ymax = 100.0        # set the extents\n",
    "    X, Y = np.mgrid[ymin:ymax:complex(ny), 0:100:complex(nx)]\n",
    "    XY = np.vstack((Y.flatten(), X.flatten())).T              # make location list for interpolation method\n",
    "    XY_list = list(map(tuple, XY))\n",
    "    x = np.zeros(100); y = np.zeros(100)     \n",
    "        \n",
    "    make_video = False\n",
    "    x[0] = 25.0; y[0] = 0.0\n",
    "    delta = 0.001\n",
    "    \n",
    "    x = np.zeros(50); y = np.zeros(50)\n",
    "    \n",
    "    plt.close()\n",
    "    # set up the plot\n",
    "    fig = plt.figure(figsize = (12,10))\n",
    "    ax = plt.gca()\n",
    "    ax.set_xlim([xmin,xmax]); ax.set_ylim([ymin,ymax])\n",
    "    ax.set_xlabel(r'$\\phi_1$'); ax.set_ylabel(r'$\\phi_2$'); ax.set_title('Gradient Descent')\n",
    "    plt.rc('xtick', labelsize=10); plt.rc('ytick', labelsize=10) \n",
    "    plt.rc('axes', titlesize=20); plt.rc('axes', labelsize=15) \n",
    "    point, = ax.plot([],[], marker=\"o\",c='white', \n",
    "                markeredgecolor = \"black\", ms=30, alpha = 0.5)\n",
    "    cbar = plt.colorbar(c1);\n",
    "    cbar.set_label('Loss Function',rotation = 270,labelpad = 20)\n",
    "    global line\n",
    "    \n",
    "    x = np.clip(x,xmin+delta,xmax-delta); y = np.clip(y,ymin+delta,ymax-delta) # ensure start is not too close to edge\n",
    "    for i in range(1,50):    \n",
    "        gradient_x = (interp((y[i-1], x[i-1]+delta)).item()-interp((y[i-1], x[i-1]-delta)).item())/(2*delta)\n",
    "        gradient_y = (interp((y[i-1]+delta, x[i-1])).item()-interp((y[i-1]-delta, x[i-1])).item())/(2*delta)\n",
    "        \n",
    "        if i > 1: # include mommentum\n",
    "            x_step = momentum*prev_x + (1.0-momentum)*(-1*r*gradient_x) # integrat momentum\n",
    "            y_step = momentum*prev_y + (1.0-momentum)*(-1*r*gradient_y) # integrat momentum\n",
    "        else: \n",
    "            x_step = -1*r*gradient_x \n",
    "            y_step = -1*r*gradient_y       \n",
    "          \n",
    "        x[i] = x[i-1] + x_step                               # step forward\n",
    "        y[i] = y[i-1] + y_step \n",
    "        \n",
    "        x[i] = np.clip(x[i],xmin+delta,xmax-delta); y[i] = np.clip(y[i],ymin+delta,ymax-delta) # ensure point is not too close to edge\n",
    "    \n",
    "        prev_x = (x[i] - x[i-1])                              # store current step for momentum on next step\n",
    "        prev_y = (y[i] - y[i-1])\n",
    "                   \n",
    "    def init():                                             # animation initialization function\n",
    "        ax = plt.gca()\n",
    "        c1 = ax.contourf(X, Y, truth, cmap = plt.cm.inferno,vmin=0,vmax=100,levels=np.linspace(0, 100, 50))\n",
    "        ax.contour(X, Y, truth, levels=10, linewidths=0.5, colors='black',linestyles='dashed') \n",
    "        \n",
    "    def update(frame):\n",
    "        N = 10\n",
    "        ax = plt.gca()\n",
    "        N = min(N,frame-1)\n",
    "        point.set_data(x[frame-N:frame],y[frame-N:frame])\n",
    "        #point.set_sizes(np.linspace(1,N,N))\n",
    "        #point.set_color(plt.cm.inferno((interp((y[frame],x[frame])).item())/100))\n",
    "    \n",
    "    ani = FuncAnimation(fig, update, frames = np.arange(1, len(x)), init_func = init, # make the animation\n",
    "        interval=100, blit=True, repeat = True)\n",
    "    \n",
    "    if make_video:                                               # set up formatting for the movie files\n",
    "        writer = animation.FFMpegWriter(fps=15, metadata=dict(artist='Me'), bitrate=1800)\n",
    "        ani.save('gradient_descent.mp4', writer=writer)\n",
    "              \n",
    "# connect the function to make the samples and plot to the widgets    \n",
    "interactive_plot = widgets.interactive_output(run_plot, {'momentum':momentum,'r':r})\n",
    "#interactive_plot.clear_output(wait = False)               # reduce flickering by delaying plot updating\n",
    "\n",
    "display(ui2, interactive_plot)                           # display the interactive plot"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "np.linspace(1,10,10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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